MULTIVALUED PERTURBATIONS FOR A CLASS OF NONLINEAR EVOLUTION-EQUATIONS

We study in this paper the initial value problem for the multivalued differential equation u' is-an-element-of - partial derivative-f(u) + G(u), where f is in MS(2) and G is a multifunction from C([0, T]; OMEGA) into the closed subsets of L2(0, Y;OMEGA), satisfying suitable regularity assumptions. As an application we prove a local existence result for the problem u'(t)2 - partial derivative- f(u(t)) + G1(t, u(t)) + G2 (t, u(t)) u(0) = u0, where G1 (resp. G2) is a lower semicontinuous (resp. upper semicontinuous, convex valued) multifunction from [0, T] x OMEGA into H. No strong compactness is required for the values of G, and G2. The differential inclusion is also investigated in the case where G comes from a multivalued regularization of g(t, x, u) and integral-t/0 a(t - s) g(s,., (s)) ds, where g(t, x, u) is a discontinuous real single-valued function defined on [0, T] x R(n) x R.